#579819

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2,4,3)(9,10)(11,13)(12,15)(14,16)(17,18)(19,21)(20,23)(22,24);(1,2,4,3)(5,6,7,8)(10,11)(14,15)(17,19,22,23)(18,20,24,21);(5,6)(7,8)(9,10,13,11)(12,15,16,14)(17,20,22,21)(18,23,24,19);(1,3,4,2)(9,11,13,10)(12,15,16,14)(17,21,22,20)(18,23,24,19);(9,12,13,16)(10,14,11,15)(17,18)(19,20)(21,23)(22,24)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=f,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 487015);